9mm NATO

I was under the impression that 9mm NATO was the same as 9mm Luger/Para but loaded to military specifications. I was browsing the SAAMI site, and noticed that NATO in a Luger/Para gun is listed as an unsafe gun/ammo combo. What gives?

In the 1950's, the US military adopted the metric system of measurement and uses metric measurements to describe ammo. However, the US commercial ammo market typically used the English "caliber" measurements when describing ammo. "Caliber" is a shorthand way of saying "hundredths (or thousandths) of an inch." For example, a fifty caliber projectile is approximately fifty one-hundredths (.50) of an inch and a 357 caliber projectile is approximately three-hundred and fifty-seven thousandths (.357) of an inch. Dimensionally, 5.56 and .223 ammo are identical, though military 5.56 ammo is typically loaded to higher pressures and velocities than commercial ammo and may, in guns with extremely tight "match" .223 chambers, be unsafe to fire.
The chambers for .223 and 5.56 weapons are not the same either. Though the AR15 design provides an extremely strong action, high pressure signs on the brass and primers, extraction failures and cycling problems may be seen when firing hot 5.56 ammo in .223-chambered rifles. Military M16s and AR15s from Colt, Bushmaster, FN, DPMS, and some others, have the M16-spec chamber and should have no trouble firing hot 5.56 ammunition.
Military M16s have slightly more headspace and have a longer throat area, compared to the SAAMI .223 chamber spec, which was originally designed for bolt-action rifles. Commercial SAAMI-specification .223 chambers have a much shorter throat or leade and less freebore than the military chamber. Shooting 5.56 Mil-Spec ammo in a SAAMI-specification chamber can increase pressure dramatically, up to an additional 15,000 psi or more.
The military chamber is often referred to as a "5.56 NATO" chamber, as that is what is usually stamped on military barrels. Some commercial AR manufacturers use the tighter ".223" (i.e., SAAMI-spec and often labeled ".223" or ".223 Remington") chamber, which provides for increased accuracy but, in self-loading rifles, less cycling reliability, especially with hot-loaded military ammo. A few AR manufacturers use an in-between chamber spec, such as the Wylde chamber. Many mis-mark their barrels too, which further complicates things. You can generally tell what sort of chamber you are dealing with by the markings, if any, on the barrel, but always check with the manufacturer to be sure.
Typical Colt Mil-Spec-type markings: C MP 5.56 NATO 1/7

I suppose I might avoid shooting NATO-spec 9mm in older/weaker pistols like P08, P35, P38, etc. But I'd have no qualms about shooting it in a good modern pistol, or a strong older design like the 1911.

So I guess that means there's a NATO chambering for 9mm with a larger throat? Wouldn't that also mean that +P or +P+ is unsafe in a standard chamber, or is NATO loaded even hotter than that?

The bullet and casing are the same. NATO standard ammo used by the military and standard in europe is close to what you see from +p loading. It means it's high pressure and offers higher velocity on the projectile.

Most military "equivalents" to civilian rounds have less case capacity and are sealed for long term storage in adverse conditions. 7.62x51 (.308), 5.56x45 (.223), 9x19 (9mm luger etc..) There are reasons why 5.56mm surplus ammo should not be shot in rifles chambered for .223 and most of the military/civilian cartridges likely follow suit in that respect. Be careful and educate yourselves. I imagine there are also reasons why in some countries that firearms are not sold or owned by the population that are made for military calibers.